Cyclic Codes and Sequences from a Class of Dembowski-Ostrom Functions

Cyclic Co des and Sequences from a Clas s of Dem b owski-Ostrom F unctions No vem b er 16, 2 018 Jinquan Luo San Ling and Chaoping Xing 0 J.Luo is with the School of Mathematics, Y angzhou Universit y , Jiangsu Province, 225009, China and with the Divisio n of Mathematics, Sc ho ol of Physics and Mathematical Sciences, Nany ang T echnological Universit y , Singapo re. S.Ling and C.Xing are with the Division of Mathematics, School of Physics and Mathematical Sciences, Na ny ang T echnological Universit y , Sing ap ore. E-mail addr esses: jqluo@ntu .edu.sg, lingsan@ntu.edu.sg, xingcp@ntu.edu.sg. 1 Abstract Let q = p n with p b e a n o dd prime. Let 0 ≤ k ≤ n − 1 and k 6 = n/ 2. In this pap er w e determine the v alue distribution o f fo llo wing exp o nen tial(c haracter) sums X x ∈ F q ζ T r n 1 ( αx p 3 k +1 + β x p k +1 ) p ( α ∈ F p m , β ∈ F q ) and X x ∈ F q ζ T r n 1 ( αx p 3 k +1 + β x p k +1 + γ x ) p ( α ∈ F p m , β , γ ∈ F q ) where T r n 1 : F q → F p and T r m 1 : F p m → F p are the canonical trace mappings and ζ p = e 2 πi p is a primitive p -th ro ot o f unit y . As applications: (1). W e determine the w eigh t distribution of the cyclic co des C 1 and C 2 o v er F p t with parity -che ck p olynomials h 2 ( x ) h 3 ( x ) and h 1 ( x ) h 2 ( x ) h 3 ( x ) resp ectiv ely where t is a divisor of d = gcd( n, k ), and h 1 ( x ), h 2 ( x ) a nd h 3 ( x ) a re the minimal p olynomials of π − 1 , π − ( p k +1) and π − ( p 3 k +1) o v er F p t resp ectiv ely for a primitiv e elemen t π of F q . (2). W e determine t he correlation distribution among a fa mily of m-sequences. Index terms: Exp onential sum, Cyclic co de, Sequence, W eight distribution, Correlation distribution 2 1 In tro ductio n Basic results on finite fields could b e found in [19]. These notatio ns are fixed throughout this pap er except for sp ecific statemen ts. • Le t p an o dd prime, p ∗ = ( − 1) p − 1 2 p , q = p n and F p , F q b e t he finite fields of order p , q resp ectiv ely . Let π b e a primitive elemen t of F q . • Le t T r j i : F p i → F p j b e the trace mapping, ζ p = exp( 2 π √ − 1 /p ) b e a p -th ro ot of unity and χ ( x ) = ζ T r n 1 ( x ) p b e the canonical additiv e c haracter on F q . • Le t k b e a p ositiv e integer, 1 ≤ k ≤ n − 1 and k / ∈ { n 4 , n 2 , 3 n 4 } . Let d = gcd( n, k ) , q 0 = p d , q ∗ 0 = ( − 1) q 0 − 1 2 q 0 and s = n/d . Let t b e a divisor of d and n 0 = n/t . • Le t m = n/ 2 (if n is ev en) a nd µ = ( − 1) m/d . Let C b e an [ l , k , d ] linear co de and A i b e the n um b er o f co dew ords in C with Hamming w eigh t i . The w eight distribution { A i } l i =0 is a n imp ortant researc h ob ject for theoretical and application in terests(see Fitzgerald and Y ucas [7], McElie ce [20], McEliece and Rumsey [21], v an der Vlugt [26], W olfmann [28] and the references therein). F o r a cyclic co de, the Hamming w eight of eac h co dew ord can b e expressed b y certain com bination o f general exponential(c haracter) sums (see F eng and Luo [5], [6], Luo and F eng [13 ], [14], Luo, T ang and W ang [15], Luo [16], v an der Vlugt [27], Y uan, Carlet and Ding [30 ], Z eng, Hu, Jia, Y ue a nd Cao [31], Zeng and Li [32]). More exactly sp eaking, let t | n , C b e the cy clic code ov er F p t with length l = q − 1 and par ity-c hec k p olynomial, h ( x ) = h 1 ( x ) · · · h u ( x ) ( u ≥ 2) where h i ( x ) ( 1 ≤ i ≤ e ) ar e distinct irreducible p olynomials in F p t [ x ] with degree e i (1 ≤ i ≤ u ) , then dim F p t C = u P i =1 e i . Let π − s i b e a zero of h i ( x ), 1 ≤ s i ≤ q − 2 (1 ≤ i ≤ u ) . Then the co dew ords in C can b e expressed b y c ( α 1 , · · · , α u ) = ( c 0 , c 1 , · · · , c l − 1 ) ( α 1 , · · · , α u ∈ F q ) 3 where c i = u P λ =1 T r n t ( α λ π is λ ) (0 ≤ i ≤ n − 1 ). Therefore t he Hamming we ight of the co dew ord c = c ( α 1 , · · · , α u ) is w H ( c ) = # { i | 0 ≤ i ≤ l − 1 , c i 6 = 0 } = l − # { i | 0 ≤ i ≤ l − 1 , c i = 0 } = l − 1 p t l − 1 X i =0 X a ∈ F p t ζ T r t 1 „ a · T r n t „ u P λ =1 α λ π is λ «« p = l − l p t − 1 p t X a ∈ F ∗ p t X x ∈ F ∗ q ζ T r n 1 ( af ( x )) p = l − l p t + p t − 1 p t − 1 p t X a ∈ F ∗ p t S ( aα 1 , · · · , aα u ) = p n − t ( p t − 1) − 1 p t X a ∈ F ∗ p t S ( aα 1 , · · · , aα u ) (1) where f ( x ) = α 1 x s 1 + α 2 x s 2 + · · · + α u x s u ∈ F q [ x ], F ∗ q = F q \{ 0 } , F ∗ p t = F p t \{ 0 } , and S ( α 1 , · · · , α u ) = X x ∈ F q ζ T r n 1 ( α 1 x s 1 + ··· + α u x s u ) p . In this w a y , the w eigh t distribution of cyclic co de C can b e deriv ed from the explicit ev aluating of the exp onential sums S ( α 1 , · · · , α u ) ( α 1 , · · · , α u ∈ F q ) . Let h 1 ( x ), h 2 ( x ) and h 3 ( x ) be the minimal p olynomials of π − 1 , π − ( p k +1) and π − ( p 3 k +1) o v er F p t resp ectiv ely . Then deg h i ( x ) = n 0 for 1 ≤ i ≤ 3 . Let C 1 and C 2 b e the cyc lic co des o v er F p t with length l = q − 1 and parit y- c hec k p olynomials h 2 ( x ) h 3 ( x ) and h 1 ( x ) h 2 ( x ) h 3 ( x ) respectiv ely . Then w e kno w that the dimensions of C 1 and C 2 o v er F p t are 2 n 0 and 3 n 0 resp ectiv ely . A Dembows ki-Ostrom function on F q is a F q -linear combination of x p i + p j with 0 ≤ i ≤ n − 1. Let f α,β ( x ) = αx p 3 k +1 + β x p k +1 for α, β ∈ F q . D efine the exp onential sums T ( α, β ) = X x ∈ F q ζ T r n 1 ( f α,β ( x ) ) p (2) 4 and for γ ∈ F q , S ( α , β , γ ) = X x ∈ F q ζ T r n 1 ( f α,β ( x )+ γ x ) p . (3) Then the w eigh t distribution of C 1 and C 2 can b e completely determined if T ( α, β ) and S ( α , β , γ ) are explicitly ev aluated. Another a pplication of S ( α, β , γ ) is to calculate the correlation distribution of corresp onding sequenc es. Let F b e a collection of p - ary m-sequences o f p erio d q − 1 defined b y F =  { a i ( t ) } q − 2 i =0 | 0 ≤ i ≤ L − 1  The c orr elation function of a i and a j for a shift τ is defined b y M i,j ( τ ) = q − 2 X λ =0 ζ a i ( λ ) − a j ( λ + τ ) p (0 ≤ τ ≤ q − 2) . In this pap er, w e will study the collection of sequences F = n a α,β =  a α,β ( π λ )  q − 2 λ =0 | α, β ∈ F q o (4) where a α,β ( π λ ) = T r n 1 ( απ λ ( p 3 k +1) + β π λ ( p k +1) + π λ ). Then the correlation function b et w een a α 1 ,β 1 and a α 2 ,β 2 b y a shift τ (0 ≤ τ ≤ q − 2) is M ( α 1 ,β 1 ) , ( α 2 ,β 2 ) ( τ ) = q − 2 P λ =0 ζ a α 1 ,β 1 ( λ ) − a α 2 ,β 2 ( λ + τ ) p = q − 2 P λ =0 ζ T r n 1 ( α 1 π λ ( p 3 k +1) + β 1 π λ ( p k +1) + π λ ) − T r n 1 ( α 2 π ( λ + τ )( p 3 k +1) + β π ( λ + τ )( p k +1) + π λ + τ ) p = S ( α ′ , β ′ , γ ′ ) − 1 (5) where α ′ = α 1 − α 2 π τ ( p 3 k +1) , β ′ = β 1 − β 2 π τ ( p k +1) , γ ′ = 1 − π τ . (6) P airs of non-binary m-sequences with few-v alued cross correlations hav e b een extensiv ely studied for sev eral decades, see Charpin [1 ], Dobbertin, Hellese th, Kumar and Martinsen [4 ], Gold [8], Helleseth [9], [10], Helleseth a nd Kumar [11], Helleseth, Lahtonen and Rosendahl [12], Kasami [1 7],[18], R osendahl [23], [24] and T rach t enberg [25], Xia and Zeng [29] a nd references therein. 5 In [31], the exp o enen tial sums T ( α, β ) and S ( α , β , γ ) for n/d o dd hav e b een ev aluated. As an a pplicatio n, the weigh t distribution to the asso ciated p -a r y cyclic co de C 2 is determined. Our pap er fo cuses on the case n/d is ev en. Moreov er, we will determine the w eight distribution of C 1 and C 2 for t | d . Mean while, the correlation distribution of sequence s in F can also b e calculated explicitly . This pap er is presen ted as follows. In Section 2 w e intro duce some prelimi- naries. In Section 3 we will study the v alue distribution o f T ( α , β ) (that is, which v alue T ( α , β ) tak es on and whic h frequency of eac h v alue) and the w eight distri- bution of C 1 . In Section 3 w e will determine the v alue distribution of S ( α , β , γ ) , the cor r elat io n distribution among the sequenc es in F , and then the w eigh t distribution of C 2 . Most length y details are presen ted in sev eral a pp endixes. The main to ols are quadratic form theory ov er o dd c haracteristic finite fields, some momen t iden tities on T ( α , β ) and a class of Artin- Sc hreier curv es on finite fields whic h w e ha v e emplo y ed in [13] a nd [1 5]. W e will fo cus our study on the o dd prime c haracteristic case and the binary case will b e in v estigated in a fo llo wing pap er. 2 Preliminaries W e follo w the not a tions in Section 1. The first machin ery to determine the v al- ues of exp o nen tial sums T ( α, β ) and S ( α, β , γ ) defined in (2) and (3) is quadratic form theory ov er F q 0 . Let H b e a n s × s symmetric matrix ov er F q 0 and r = rank H . Then there exists M ∈ GL s ( F q 0 ) suc h that H ′ = M H M T is diagonal and H ′ = diag ( a 1 , · · · , a r , 0 , · · · , 0) where a i ∈ F ∗ q 0 (1 ≤ i ≤ r ). Let ∆ = a 1 · · · a r (w e assume ∆ = 1 when r = 0) and η 0 b e the quadratic (m ultiplicativ e) ch aracter of F q 0 . Then η 0 (∆) is an inv arian t of H under the conjugate action of M ∈ GL s ( F q 0 ). F o r t he quadratic form F : F s q 0 → F q 0 , F ( x ) = X H X T ( X = ( x 1 , · · · , x s ) ∈ F s q 0 ) , (7) w e ha v e the following result(see [13], Lemma 1 ) . L emma 1. (i). F or the quadr atic f o rm F = X H X T define d in (7), we have X X ∈ F s q 0 ζ T r d 1 ( F ( X )) p = ( η 0 (∆) q s − r / 2 0 if q 0 ≡ 1 (mo d 4) , i r η 0 (∆) q 0 s − r / 2 if q 0 ≡ 3 (mo d 4) . 6 (ii). F or A = ( a 1 , · · · , a s ) ∈ F s q 0 , if 2 Y H + A = 0 has solution Y = B ∈ F s q 0 , then P X ∈ F s q 0 ζ T r d 1 ( F ( X )+ AX T ) p = ζ c p P X ∈ F s q 0 ζ T r d 1 ( F ( X )) p wher e c = − T r d 1  B H B T  = 1 2 T r d 1  AB T  ∈ F p . Otherwise P X ∈ F m p ζ T r d 1 ( F ( X )+ AX T ) p = 0 . In this corresp ondence w e alw a ys assume d = gcd( n, k ). Then the field F q is a v ector space o v er F q 0 with dimension s . W e fix a basis v 1 , · · · , v s of F q o v er F q 0 . Then eac h x ∈ F q can b e uniquely expressed as x = x 1 v 1 + · · · + x s v s ( x i ∈ F q 0 ) . Th us w e hav e the following F q 0 -linear isomorphism: F q ∼ − → F s q 0 , x = x 1 v 1 + · · · + x s v s 7→ X = ( x 1 , · · · , x s ) . With this isomorphism, a f unction f : F q → F q 0 induces a f unction F : F s q 0 → F q 0 where for X = ( x 1 , · · · , x s ) ∈ F s q 0 , F ( X ) = f ( x ) with x = x 1 v 1 + · · · + x s v s . In this w a y , function f ( x ) = T r n d ( γ x ) f or γ ∈ F q induces a linear form F ( X ) = T r n d ( γ x ) = s X i =1 T r n d ( γ v i ) x i = A γ X T (8) where A γ = (T r n d ( γ v 1 ) , · · · , T r n d ( γ v s )) , and f α,β ( x ) = T r n d ( αx p 3 k +1 + β x p k +1 ) for α, β ∈ F q induces a quadratic form F α,β ( X ) = T r n d ( αx p 3 k +1 + β x p k +1 ) = T r n d α s X i =1 x i v p 3 k i ! s X i =1 x i v i ! + β s X i =1 x i v p k i ! s X i =1 x i v i !! = s X i,j =1 T r n d  αv p 3 k i v j + β v p k i v j  x i x j = X H α,β X T (9) where H α,β = ( h ij ) s × s and h ij = 1 2 T r n d  α  v p 3 k i v j + v i v p 3 k j  + β  v p k i v j + v i v p k j  for 1 ≤ i, j ≤ s. 7 F ro m Lemma 1, in order to determine the v alues of T ( α, β ) = X x ∈ F q ζ T r n 1 ( αx p 3 k +1 + β x p k +1 ) p = X X ∈ F s q 0 ζ T r d 1 ( X H α,β X T ) p and S ( α , β , γ ) = X x ∈ F q ζ T r n 1 ( αx p 3 k +1 + β x p k +1 + γ x ) p = X X ∈ F m p ζ T r d 1 ( X H α,β X T + A γ X T ) p ( α, β , γ ∈ F q ) , w e need to determine the rank of H α,β o v er F q 0 and the solv ability of F q 0 -linear equation 2 X H α,β + A γ = 0. Define d ′ = gcd ( n, 2 k ). Then an easy observ atio n shows d ′ =  2 d, if n/d is ev en; d, otherwise. (10) No w w e could determine the p ossible ranks o f H α,β . L emma 2. F or α, β ∈ F q and ( α, β ) 6 = { (0 , 0 ) } , let r α,β b e the r an k of H α,β . Then we have (i). if d ′ = d , then the p ossible values of r α,β ar e s , s − 1 , s − 2 . (ii). if d ′ = 2 d , then the p ossible values of r α,β ar e s , s − 2 , s − 4 , s − 6 . Pr o of. F or (i), see [3 1 ]. F o r (ii), see App endix A. In o r der to determine the v alue distribution of T ( α, β ) for α, β ∈ F q , we need the followin g result on momen ts of T ( α , β ). L emma 3. F or the exp onential sum T ( α, β ) , ( i ) . P α,β ∈ F q T ( α, β ) = p 2 n ; ( ii ) . P α,β ∈ F q T ( α, β ) 2 =      p 2 n if d ′ = d and p d ≡ 3 (mo d 4) , (2 p n − 1) · p 2 n if d ′ = d and p d ≡ 1 (mo d 4 ) , ( p n + d + p n − p d ) · p 2 n if d ′ = 2 d ; ( iii ) . if d ′ = 2 d, then P α,β ∈ F q T ( α, β ) 3 = ( p n +3 d + p n − p 3 d ) · p 2 n . 8 Pr o of. see App endix A. In the case d ′ = 2 d , w e could determine the explicit v alues of T ( α, β ). T o this end we will study a class of Artin- Sc hreier curv es. A similar tec hnique has b een applied in Coulter [2 ], Theorem 6 .1. L emma 4. Supp ose ( α, β ) ∈ ( F q × F q )  { 0 , 0 } and d ′ = 2 d . L et N b e the numb er of F q -r a tional (affine) p oints on the curve αx p 3 k +1 + β x p k +1 = y p d − y . (11) Then N = q + ( p d − 1) · T ( α , β ) . Pr o of. see App endix A. No w w e giv e an explicit ev a luation of T ( α , β ) in the case d ′ = 2 d . L emma 5. Assumptions as in L emma 4 and let n = 2 m , then T ( α, β ) =            µp m , if r α,β = s − µp m + d , if r α,β = s − 2 µp m +2 d , if r α,β = s − 4 − µp m +3 d , if r α,β = s − 6 . wher e µ = ( − 1) m/d . Pr o of. Consider the F q -rational (affine) p oints on the Art in- Sc hreier curve in Lemma 4. It is easy to v erify that (0 , y ) with y ∈ F p d are exactly the p oints on the curv e with x = 0. If ( x, y ) with x 6 = 0 is a p o in t on this curv e, then so are ( tx, t p d +1 y ) with t p 2 d − 1 = 1 (not e that p 3 k + 1 ≡ p k + 1 ≡ p d + 1 (mo d p 2 d − 1) since 3 k /d and k /d are b ot h o dd b y (10)). In to t al, w e hav e q + ( p d − 1) T ( α , β ) = N ≡ p d (mo d p 2 d − 1) whic h yields T ( α, β ) ≡ 1 (mo d p d + 1) . W e o nly consider the case r α,β = s and m/d is o dd. The other cases are similar. In this case T ( α, β ) = ± p m . Assume T ( α , β ) = p m . Then p d + 1 | p m − 1 whic h contradicts to m/d is o dd. Therefore T ( α , β ) = − p m . 9 R emark . (i). Our treatmen t impro v e the tec hnique in [2], [3], in whic h the case ( p, d ) = (3 , 1) is dealt with in a different manner. (ii). Applying Lemma 5 to Lemma 4, we could determine the n um b er of rational p oints on the curv e (11 ). 3 Exp onen tial Sums T ( α , β ) and Cyclic Co de C 1 Define N i = { ( α, β ) ∈ F q × F q \{ (0 , 0) } | r α,β = s − i } . Then n i =   N i   for i = 0 , 2 , 4 , 6. According to the p ossible v alues of T ( α , β ) giv en by Lemma 1 (setting F ( X ) = X H α,β X T = T r n d ( αx p 3 k +1 + β x p k +1 )), w e define that for ε = ± 1, N ε,i =      n ( α, β ) ∈ F 2 q \{ (0 , 0) }    T ( α, β ) = εp n + id 2 o if n + id is ev en , n ( α, β ) ∈ F 2 q \{ (0 , 0) }    T ( α, β ) = ε √ p ∗ p n + id − 1 2 o if n + id is o dd where p ∗ = ( − 1) p − 1 2 p and n ε,i = | N ε,i | . Recall q ∗ 0 = ( − 1) q 0 − 1 2 q 0 . In this section w e prov e the follo wing results. The or em 1. The value distribution of the m ulti-set { T ( α , β ) | α, β ∈ F q } is shown as fol lowing. (i). F o r the c ase d ′ = d , values multiplicity √ q ∗ 0 q 0 s − 1 2 , − √ q ∗ 0 q 0 s − 1 2 1 2 p 2 d ( p n − p n − d − p n − 2 d + 1) ( p n − 1) / ( p 2 d − 1) p n + d 2 1 2 p n − d 2 ( p n − d 2 + 1) ( p n − 1) − p n + d 2 1 2 p n − d 2 ( p n − d 2 − 1)( p n − 1) √ q 0 ∗ q 0 s +1 2 , − √ q 0 ∗ q 0 s +1 2 1 2 ( p n − 1)( p n − d − 1) / ( p 2 d − 1) p n 1 (ii). F or the c ase d ′ = 2 d , 10 values multiplicity µp m ( p n − 1) ( p n +6 d − p n +4 d − p n + d + µp m +5 d − µp m +4 d + p 6 d ) ( p d +1)( p 2 d − 1)( p 3 d +1) − µp m + d ( p n − 1)( p n +3 d + p n +2 d − p n − p n − d − p n − 2 d − µp m +3 d + µp m + p 3 d ) ( p d +1) 2 ( p 2 d − 1) µp m +2 d ( p m − d + µ )( p m + d + p m − p m − 2 d − µp d )( p n − 1) ( p d +1) 3 ( p d − 1) − µp m +3 d ( p m − 2 d − µ )( p m − d + µ )( p n − 1) ( p d +1)( p 2 d − 1)( p 3 d +1) p n 1 wher e µ = ( − 1) m/d . Pr o of. see [31] for (i) and App endix B for (ii). R emark . (i). In the c ase k ∈  n 4 , n 2 , 3 n 4  , the exp o nen tial s um T ( α, β ) = P x ∈ F q χ (( α p k + β ) x p k +1 ) ha s b een extensiv ely studied, for example, see [2], [19]. (ii). In t he case k ∈  n 6 , 5 n 6  , the exp onen tial sum T ( α , β ) = P x ∈ F q χ ( αx p m +1 + β x p k +1 ) is a sp ecial case in [15]. In the case k ∈ { n 3 , 2 n 3 } , the exp onen tial sum T ( α , β ) = P x ∈ F q χ ( αx 2 + β x p k +1 ) is a sp ecial case in [13]. Recall that t is a divisor of d and C 1 is the cyclic co de o v er F p t with parit y- c hec k p olynomial h 2 ( x ) h 3 ( x ) where h 2 ( x ) and h 3 ( x ) are the minimal p olynomials of π − ( p 3 k +1) and π − ( p k +1) , resp ective ly . The or em 2. Supp ose that 1 ≤ k ≤ n − 1 and k / ∈  n 4 , n 2 , 3 n 4  . Then the weight distribution { A 0 , A 1 , · · · , A q − 1 } of the cycli c c o de C 1 over F p t ( p ≥ 3 ) with length q − 1 is shown as fol lo w ing. (i). F or the c ase d = d ′ and d/ t ar e b oth o dd, then dim F p t C 1 = 2 n 0 and A i = 0 exc ept for i A i ( p t − 1)( p n − t − p n + d 2 − t ) 1 2 p n − d 2 ( p n − d 2 + 1) ( p n − 1) ( p t − 1) p n − t ( p n − 1)( p n − p n − d + 1) ( p t − 1)( p n − t + p n + d 2 − t ) 1 2 p n − d 2 ( p n − d 2 − 1)( p n − 1) 0 1 11 (ii). F or the c ase d = d ′ and d/t is even, then dim F p t C 1 = 2 n 0 and A i = 0 exc ept for i A i ( p t − 1)( p n − t − p n 2 + d − t ) 1 2 ( p n − 1)( p n − d − 1)  ( p 2 d − 1) ( p t − 1)( p n − t − p n + d 2 − t ) 1 2 p n − d 2 ( p n − d 2 + 1) ( p n − 1) ( p t − 1)( p n − t − p n 2 − t ) 1 2 p 2 d ( p n − p n − d − p n − 2 d + 1) ( p n − 1)  ( p 2 d − 1) ( p t − 1)( p n − t + p n 2 − t ) 1 2 p 2 d ( p n − p n − d − p n − 2 d + 1) ( p n − 1)  ( p 2 d − 1) ( p t − 1)( p n − t + p n + d 2 − t ) 1 2 p n − d 2 ( p n − d 2 − 1)( p n − 1) ( p t − 1)( p n − t + p n 2 + d − t ) 1 2 ( p n − 1)( p n − d − 1)  ( p 2 d − 1) 0 1 (iii). F or the c ase d ′ = 2 d and k / ∈  n 6 , 5 n 6  , then dim F p t C 1 = 2 n 0 and A i = 0 exc ept for i A i ( p t − 1)( p n − t + µp m +3 d − t ) ( p m − 2 d − µ )( p m − d + µ )( p n − 1) ( p d +1)( p 2 d − 1)( p 3 d +1) ( p t − 1)( p n − t − µp m +2 d − t ) ( p m − d + µ )( p m + d + p m − p m − 2 d − µp d )( p n − 1) ( p d +1) 3 ( p d − 1) ( p t − 1)( p n − t + µp m + d − t ) ( p n − 1)( p n +3 d + p n +2 d − p n − p n − d − p n − 2 d − µp m +3 d + µp m + p 3 d ) ( p d +1) 2 ( p 2 d − 1) ( p t − 1)( p n − t − µp m − t ) ( p n − 1)( p n +6 d − p n +4 d − p n + d + µp m +5 d µp m +4 d + p 6 d ) ( p d +1)( p 2 d − 1)( p 3 d +1) 0 1 (iii). F or the c ase d ′ = 2 d and k ∈  n 6 , 5 n 6  , then dim F p t C 1 = 3 n 0 / 2 and A i = 0 exc ept for i A i ( p t − 1)( p n − t + p 5 n 6 − t ) ( p n − 1) / ( p n 6 + 1) ( p t − 1)( p n − t − p 2 n 3 − t ) p n 6 ( p n 3 + 1) ( p n − 1) / ( p n 6 + 1) ( p t − 1)( p n − t + p n 2 − t ) p n 2 ( p n 2 − 1)( p 2 n 3 − 1) / ( p n 6 + 1) 0 1 12 Pr o of. see App endix B. R emark . ( 1 ). In the case d = d ′ . Since gcd( p n − 1 , p k + 1) = 2, the first l ′ = q − 1 2 co ordinates of eac h co dew ord of C 1 form a cyclic co de C ′ 1 o v er F p t with length l ′ and dimension 2 n 0 . Let ( A ′ 0 , · · · , A ′ l ′ ) b e the weigh t distribution of C ′ 1 , then A ′ i = A 2 i (0 ≤ i ≤ l ′ ). (2). In the case d ′ = 2 d . Since gcd( p n − 1 , p k + 1) = p d + 1, the first l ′ = q − 1 p d +1 co ordinates of eac h co dew ord of C 1 form a cyclic co de C ′ 1 o v er F p t with length l ′ and dimension 2 n 0 . Let ( A ′ 0 , · · · , A ′ l ′ ) b e the weigh t distribution of C ′ 1 , then A ′ i = A ( p d +1) i (0 ≤ i ≤ l ′ ). (3). If k ∈  n 4 , n 2 , 3 n 4  , then π − ( p 3 k +1) p k = π − ( p k +1) and the dual code of C 1 has only one zero. This sp ecial case is trivial. 4 Results on Correlation Di stribut ion o f Se quences and Cycli c Co d e C 2 Recall φ α,β ( x ) in the pro of of Lemma 2 and N i,ε in the pro of o f Theorem 1. Finally w e will determine the v alue distribution of S ( α, β , γ ) , the correlation distribution among sequences in F defined in (4 ) and the w eigh t distribution of C 2 defined in Section 1. The following result will play an impo r tan t ro le. L emma 6. Assume t b e a diviso r of d . F or any a ∈ F p t and any ( α, β ) ∈ N i,ε with ε = ± 1 , then the numb er of elemen ts γ ∈ F q satisfying (i). φ α,β ( x ) + γ = 0 is solv a ble(cho ose one solution, say x 0 ), (ii). T r n t ( αx p 3 k +1 0 + β x p k +1 0 ) = a is              p n − id − t if s − i and d/t ar e b oth o dd , and a = 0 , p n − id − t + ε η ′ ( a ) p n − id − t 2 if s − i and d/t ar e b oth o dd , and a 6 = 0 , p n − id − t + ε ( p t − 1) p n − id 2 − t if s − i or d/t is even , and a = 0 , p n − id − t − εp n − id 2 − t if s − i or d/t is even , and a 6 = 0 . wher e η ′ is the quadr atic ( multiplic ative) char acter o n F p t . 13 Pr o of. see App endix C. Let p ∗ = ( − 1) p − 1 2 p and  · p  b e the Legendre sym b ol. W e are no w ready to giv e the v alue distribution o f S ( α , β , γ ). The or em 3. The value di s tribution of the multi-set  S ( α , β , γ )   α ∈ F p m , ( β , γ ) ∈ F 2 q  is shown as fol lowing. (i). I f d ′ = d is o dd (that is n is o dd), then values multiplicity ε √ p ∗ p n − 1 2 1 2 p n +2 d − 1 ( p n − p n − d − p n − 2 d +1)( p n − 1) p 2 d − 1 εζ j p √ p ∗ p n − 1 2 1 2 p 2 d ( p n − 1 + ε  − j p  p n − 1 2 ) ( p n − p n − d − p n − 2 d +1)( p n − 1) p 2 d − 1 εp n + d 2 1 2 p n − d − 1 ( p n − d 2 + ε ( p − 1) ) ( p n − d 2 + ε ) ( p n − 1) εζ j p p n + d 2 1 2 p n − d − 1 ( p n − d 2 − ε )( p n − d 2 + ε ) ( p n − 1) ε √ p ∗ p n +2 d − 1 2 1 2 p n − 2 d − 1 ( p n − 1)( p n − d − 1) p 2 d − 1 εζ j p √ p ∗ p n +2 d − 1 2 1 2 ( p n − 2 d − 1 + ε  − j p  p n − 2 d − 1 2 ) ( p n − 1)( p n − d − 1) p 2 d − 1 0 ( p n − 1)( p 2 n − d − p 2 n − 2 d + p 2 n − 3 d − p n − 2 d + 1) p n 1 wher e ε = ± 1 , 1 ≤ j ≤ p − 1 . (ii). I f d ′ = d is even, then values multiplicity εp n 2 1 2 p 2 d ( p n − 1 + ε ( p − 1) p n 2 − 1 ) ( p n − p n − d − p n − 2 d +1)( p n − 1) p 2 d − 1 εζ j p p n 2 1 2 p 2 d ( p n − 1 − εp n 2 − 1 ) ( p n − p n − d − p n − 2 d +1)( p n − 1) p 2 d − 1 εp n + d 2 1 2 p n − d − 1 ( p n − d 2 + ε ( p − 1) ) ( p n − d 2 + ε ) ( p n − 1) εζ j p p n + d 2 1 2 p n − d − 1 ( p n − d 2 − ε )( p n − d 2 + ε ) ( p n − 1) εp n +2 d 2 1 2 ( p n − 2 d − 1 + ε ( p − 1) p n − 2 d 2 − 1 ) ( p n − 1)( p n − d − 1) p 2 d − 1 εζ j p p n +2 d 2 1 2 ( p n − 2 d − 1 − εp n − 2 d 2 − 1 ) ( p n − 1)( p n − d − 1) p 2 d − 1 0 ( p n − 1)( p 2 n − d − p 2 n − 2 d + p 2 n − 3 d − p n − 2 d + 1) p n 1 14 wher e ε = ± 1 , 1 ≤ j ≤ p − 1 . (iii). F or the c ase d ′ = 2 d , values multiplicity µp m ( p n − 1)( p n − 1 + µ ( p − 1) p m − 1 )( p n +6 d − p n +4 d − p n + d + µp m +5 d − µp m +4 d + p 6 d ) ( p d +1)( p 2 d − 1)( p 3 d +1) µζ j p p m ( p n − 1)( p n − 1 − µp m − 1 )( p n +6 d − p n +4 d − p n + d + µp m +5 d − µp m +4 d + p 6 d ) ( p d +1)( p 2 d − 1)( p 3 d +1) − µp m + d ( p n − 2 d − 1 − µ ( p − 1) p m − d − 1 )( p n − 1)( p n +3 d + p n +2 d − p n − p n − d − p n − 2 d − µp m +3 d + µp m + p 3 d ) ( p d +1) 2 ( p 2 d − 1) − µζ j p p m + d ( p n − 2 d − 1 + µp m − d − 1 )( p n − 1)( p n +3 d + p n +2 d − p n − p n − d − p n − 2 d − µp m +3 d + µp m + p 3 d ) ( p d +1) 2 ( p 2 d − 1) µp m +2 d ( p m − d + µ )( p m + d + p m − p m − 2 d − µp d )( p n − 4 d − 1 + µ ( p − 1) p m − 2 d − 1 )( p n − 1) ( p d +1) 2 ( p 2 d − 1) µζ j p p m +2 d ( p m − d + µ )( p m + d + p m − p m − 2 d − µp d )( p n − 4 d − 1 − µp m − 2 d − 1 )( p n − 1) ( p d +1) 2 ( p 2 d − 1) − µp m +3 d ( p m − 2 d − µ )( p m − d + µ )( p n − 6 d − 1 − µ ( p − 1) p m − 3 d − 1 )( p n − 1) ( p d +1)( p 2 d − 1)( p 3 d +1) − µζ j p p m +3 d ( p m − 2 d − µ )( p m − d + µ )( p n − 6 d − 1 + µp m − 3 d − 1 )( p n − 1) ( p d +1)( p 2 d − 1)( p 3 d +1) 0 ( p n − 1)  1 − µp 3 m − d − µp 3 m − 8 d + p n − d + p 2 n + p 2 n − 9 d + µp 3 m − 3 d − µp 3 m − 5 d − p n − 4 d − p n − 6 d p d +1  p n 1 for 1 ≤ j ≤ p − 1 . Pr o of. F or (i) and (ii), see [31]. F or (iii), see App endix C. In order to giv e the correlation distribution among t he sequenc es in F , we need an easy observ ation. L emma 7. F or any given γ ∈ F ∗ q , wh e n ( α, β ) runs thr ough F q × F q , the distri- bution o f S ( α, β , γ ) is the same as S ( α , β , 1) . As a conseque nce of Theorem 1, Theorem 3 and Lemma 7, w e could giv e the correlation distribution amidst the sequences in F . The or em 4. L et 1 ≤ k ≤ n − 1 and k / ∈  n 6 , n 4 , n 2 , 3 n 4 , 5 n 6  . The c ol le ction F define d in (4) is a family of p 2 n p -ary se quenc e s with p erio d q − 1 . Its c orr elation distribution i s given a s fol lows. (i). F o r the c ase d ′ = d is o dd (that is n is o dd), then 15 values multiplicity ε √ p ∗ p n − 1 2 − 1 1 2 p 2 n +2 d ( p 2 n − 1 − 2 p n − 1 + 1) p n − p n − d − p n − 2 d +1 p 2 d − 1 εζ j p √ p ∗ p n − 1 2 − 1 1 2 p 2 n +2 d ( p n − 1 + ε  − j p  p n − 1 2 )( p n − 2) p n − p n − d − p n − 2 d +1 p 2 d − 1 εp n + d 2 − 1 1 2 p 5 n − d 2 ( p n − d 2 + ε )  p n − d 2 − 1 ( p n − d 2 + ε ( p − 1) ) ( p n − 2) + 1  εζ j p p n + d 2 − 1 1 2 p 3 n − d − 1 ( p n − d 2 − ε )( p n − d 2 + ε ) ( p n − 2) ε √ p ∗ p n +2 d − 1 2 − 1 1 2 p 2 n ( p 2 n − 2 d − 1 − 2 p n − 2 d − 1 + 1) p n − d − 1 p 2 d − 1 εζ j p √ p ∗ p n +2 d − 1 2 − 1 1 2 p 2 n ( p n − 2 d − 1 + ε  − j p  p n − 2 d − 1 2 )( p n − 2) p n − d − 1 p 2 d − 1 − 1 p 2 n ( p n − 2)( p 2 n − d − p 2 n − 2 d + p 2 n − 3 d − p n − 2 d + 1) p n − 1 p 2 n wher e ε = ± 1 , 1 ≤ j ≤ p − 1 . (ii). F or the c ase d ′ = d is even, then values multiplicity εp n 2 − 1 1 2 p 2 n +2 d  ( p n − 1 + ε ( p − 1) p n 2 − 1 )( p n − 2) + 1  p n − p n − d − p n − 2 d +1 p 2 d − 1 εζ j p p n 2 − 1 1 2 p 2 n +2 d ( p n − 1 − εp n 2 − 1 )( p n − 2) p n − p n − d − p n − 2 d +1 p 2 d − 1 εp n + d 2 − 1 1 2 p 5 n − d 2 ( p n − d 2 + ε )  p n − d 2 ( p n − d 2 + ε ( p − 1) ) ( p n − 2) + 1  εζ j p p n + d 2 − 1 1 2 p 5 n − d − 1 ( p n − d 2 − ε )( p n − d 2 + ε ) ( p n − 2) εp n +2 d 2 − 1 1 2 p 2 n  p n − 2 d − 1 + ε ( p − 1) p n − 2 d 2 − 1  ( p n − 2) + 1  p n − d − 1 p 2 d − 1 εζ j p p n +2 d 2 − 1 1 2 p 2 n ( p n − 2 d − 1 − εp n − 2 d 2 − 1 )( p n − 2)( p n − d − 1) / ( p 2 d − 1) − 1 p 2 n ( p n − 2)( p 2 n − d − p 2 n − 2 d + p 2 n − 3 d − p n − 2 d + 1) p n − 1 p 2 n wher e ε = ± 1 , 1 ≤ j ≤ p − 1 . (iii). F or the c ase d ′ = 2 d , 16 values multiplicity µp m − 1 p 2 n ( ( p n − 2)( p n − 1 + µ ( p − 1) p m − 1 )+1 )( p n +6 d − p n +4 d − p n + d + µp m +5 d − µp m +4 d + p 6 d ) ( p d +1)( p 2 d − 1)( p 3 d +1) µζ j p p m − 1 p 2 n ( p n − 2)( p n − 1 − µp m − 1 )( p n +6 d − p n +4 d − p n + d + µp m +5 d − µp m +4 d + p 6 d ) ( p d +1)( p 2 d − 1)( p 3 d +1) − µp m + d − 1 p 2 n ( ( p n − 2 d − 1 − µ ( p − 1) p m − d − 1 )( p n − 2)+1 ) ( p n +3 d + p n +2 d − p n − p n − d − p n − 2 d − µp m +3 d + µp m + p 3 d ) ( p d +1) 2 ( p 2 d − 1) − µζ j p p m + d − 1 p 2 n ( p n − 2 d − 1 + µp m − d − 1 )( p n − 2)( p n +3 d + p n +2 d − p n − p n − d − p n − 2 d − µp m +3 d + µp m + p 3 d ) ( p d +1) 2 ( p 2 d − 1) µp m +2 d − 1 p 2 n ( ( p m − d + µ )( p n − 2)+1 ) ( p m + d + p m − p m − 2 d − µp d )( p n − 4 d − 1 + µ ( p − 1) p m − 2 d − 1 ) ( p d +1) 2 ( p 2 d − 1) µζ j p p m +2 d − 1 p 2 n ( p m − d + µ )( p m + d + p m − p m − 2 d − µp d )( p n − 4 d − 1 − µp m − 2 d − 1 )( p n − 2) ( p d +1) 2 ( p 2 d − 1) − µp m +3 d − 1 p 2 n ( ( p m − 2 d − µ )( p n − 2)+1 ) ( p m − d + µ )( p n − 6 d − 1 − µ ( p − 1) p m − 3 d − 1 ) ( p d +1)( p 2 d − 1)( p 3 d +1) − µζ j p p m +3 d − 1 p 2 n ( p m − 2 d − µ )( p m − d + µ )( p n − 6 d − 1 + µp m − 3 d − 1 )( p n − 2) ( p d +1)( p 2 d − 1)( p 3 d +1) − 1 p 2 n ( p n − 2)  1 − µp 3 m − d − µp 3 m − 8 d + p n − d + p 2 n + p 2 n − 9 d + µp 3 m − 3 d − µp 3 m − 5 d − p n − 4 d − p n − 6 d p d +1  p n − 1 p 2 n for 1 ≤ j ≤ p − 1 a n d µ = ( − 1) m/d . Recall that C 2 is the cyclic co de ov er F p t with parity-c hec k p olynomial h 1 ( x ) h 2 ( x ) h 3 ( x ) where h 1 ( x ), h 2 ( x ) and h 3 ( x ) are the minimal p olynomials of π − 1 , π − ( p k +1) and π − ( p 3 k +1) resp ectiv ely . Here w e are ready to determine the w eigh t distribution of C 2 . The or em 5. F or n ≥ 3 , k / ∈  n 6 , n 4 , n 2 , 3 n 4 , 5 n 6  , the w e i g ht distribution { A 0 , A 1 , · · · , A q − 1 } of the cyclic c o de C 2 over F p t ( p ≥ 3 ) with length q − 1 and dim F p t C 1 = 3 n 0 is shown as fol lows: (i). the c ase d ′ = d and d / t is o dd, 17 i A i ( p t − 1) p n − t − p n +2 d − t 2 1 2 p n − 2 d − t 2 ( p t − 1)( p n − 2 d − t 2 + 1) ( p n − d − 1)( p n − 1) p 2 d − 1 ( p t − 1)( p n − d − p n + d − 2 t 2 ) 1 2 p n − d − t ( p n − d 2 + 1) ( p n − d 2 + p t − 1)( p n − 1) ( p t − 1) p n − t − p n + d − 2 t 2 1 2 p n − d − t ( p t − 1)( p n − d − 1)( p n − 1) ( p t − 1) p n − t − p n − t 2 1 2 p n − t 2 +2 d ( p t − 1)( p n − t 2 + 1) ( p n − p n − d − p n − 2 d + 1) p n − 1 p 2 d − 1 ( p t − 1) p n − t ( p n − 1)( p 2 n − t + p 2 n − d − p 2 n − d − t − p 2 n − 2 d + p 2 n − 3 d − p 2 n − 3 d − t + p n − t − p n − 2 d + p n − 2 d − t + 1) ( p t − 1) p n − t + p n − t 2 1 2 p n − t 2 +2 d ( p t − 1)( p n − t 2 − 1)( p n − p n − d − p n − 2 d + 1) p n − 1 p 2 d − 1 ( p t − 1) p n − t + p n + d − 2 t 2 1 2 p n − d − t ( p t − 1)( p n − d − 1)( p n − 1) ( p t − 1)( p n − d + p n + d − 2 t 2 ) 1 2 p n − d − t ( p n − d 2 − 1)( p n − d 2 − p t + 1) ( p n − 1) ( p t − 1) p n − t + p n +2 d − t 2 1 2 p n − 2 d − t 2 ( p t − 1)( p n − 2 d − t 2 − 1) ( p n − d − 1)( p n − 1) p 2 d − 1 0 1 (ii). the c ase d ′ = d and d/t is even, 18 i A i ( p t − 1)( p n − t − p n 2 + d − t ) 1 2 p n 2 − d − t ( p n 2 − d + p t − 1)( p n − d − 1) p n − 1 p 2 d − 1 ( p t − 1) p n − t − p n 2 + d − t 1 2 p n 2 − d − t ( p t − 1)( p n 2 − d + 1) ( p n − d − 1) p n − 1 p 2 d − 1 ( p t − 1)( p n − t − p n + d 2 − t ) 1 2 p n − d − t ( p n − d 2 + 1) ( p n − d 2 + p t − 1)( p n − 1) ( p t − 1) p n − t − p n + d 2 − t 1 2 p n − d − t ( p t − 1)( p n − d − 1)( p n − 1) ( p t − 1)( p n − t − p n 2 − t ) 1 2 p n 2 +2 d − t ( p n 2 + p t − 1)( p n − p n − d − p n − 2 d + 1) p n − 1 p 2 d − 1 ( p t − 1) p n − t − p n 2 − t 1 2 p n 2 +2 d − t ( p t − 1)( p n 2 + 1) ( p n − p n − d − p n − 2 d + 1) p n − 1 p 2 d − 1 ( p t − 1) p n − t ( p n − 1)( p 2 n − d − p 2 n − 2 d + p 2 n − 3 d − p n − 2 d + 1) ( p t − 1) p n − t + p n 2 − t 1 2 p n 2 +2 d − t ( p t − 1)( p n 2 − 1)( p n − p n − d − p n − 2 d + 1) p n − 1 p 2 d − 1 ( p t − 1)( p n − t + p n 2 − t ) 1 2 p n 2 +2 d − t ( p n 2 − p t + 1) ( p n − p n − d − p n − 2 d + 1) p m − 1 p 2 d − 1 ( p t − 1) p n − t + p n + d 2 − t 1 2 p n − d − t ( p t − 1)( p n − d − 1)( p n − 1) ( p t − 1)( p n − t + p n + d 2 − t ) 1 2 p n − d − t ( p n − d 2 − 1)( p n − d 2 − p t + 1) ( p n − 1) ( p t − 1) p n − t + p n 2 + d − t 1 2 p n 2 − d − t ( p t − 1)( p n 2 − d − 1)( p n − d − 1) p n − 1 p 2 d − 1 ( p t − 1)( p n − t + p n 2 − d − t ) 1 2 p n 2 + d − t ( p n 2 − d − p t + 1) ( p n − d − 1) p n − 1 p 2 d − 1 0 1 (iii). the c ase d ′ = 2 d , 19 i A i ( p t − 1) ( p n − t − µp m − t ) ( p n − t + µ ( p t − 1) p m − t ) ( p n − 1) ( p n +6 d − p n +4 d − p n + d + µp m +5 d − µp m +4 d + p 6 d ) ( p d +1)( p 2 d − 1)( p 3 d +1) ( p t − 1) p n − t + µp m − t ( p t − 1)( p n − t − µp m − t )( p n − 1) ( p n +6 d − p n +4 d − p n + d + µp m +5 d − µp m +4 d + p 6 d ) ( p d +1)( p 2 d − 1)( p 3 d +1) ( p t − 1)  p n − t + µp m + d − t  ( p n − 2 d − t − µ ( p t − 1) p m − d − t ) ( p n − 1)( p n +3 d + p n +2 d − p n − p n − d − p n − 2 d − µp m +3 d + µp m + p 3 d ) ( p d +1) 2 ( p 2 d − 1) ( p t − 1) p n − t − µp m + d − t ( p t − 1) ( p n − 2 d − t + µp m − d − t ) ( p n − 1)( p n +3 d + p n +2 d − p n − p n − d − p n − 2 d − µp m +3 d + µp m + p 3 d ) ( p d +1) 2 ( p 2 d − 1) ( p t − 1)  p n − t − µp m +2 d − t  ( p n − 4 d − t + µ ( p t − 1) p m − 2 d − t )( p m − d + µ )( p m + d + p m − p m − 2 d − µp d )( p n − 1) ( p d +1) 3 ( p d − 1) ( p t − 1) p n − t + µp m +2 d − t ( p t − 1)( p n − 4 d − t − µp m − 2 d − t )( p m − d + µ )( p m + d + p m − p m − 2 d − µp d )( p n − 1) ( p d +1) 3 ( p d − 1) ( p t − 1)  p n − t + µp m +3 d − t  ( p n − 6 d − t − µ ( p t − 1) p m − 3 d − t ) ( p m − 2 d − µ )( p m − d + µ )( p n − 1) ( p d +1)( p 2 d − 1)( p 3 d +1) ( p t − 1) p n − t − µp m +3 d − t ( p t − 1) ( p n − 6 d − t + µp m − 3 d − t ) ( p m − 2 d − µ )( p m − d + µ )( p n − 1) ( p d +1)( p 2 d − 1)( p 3 d +1) ( p t − 1) p n − t ( p 2 n + p 2 n − 9 d − µp 3 m + µp 3 m − d + µp 3 m − 3 d − µp 3 m − 5 d − µp 3 m − 7 d + µp 3 m − 8 d + p n − p n − d − p n − 4 d − p n − 6 d + p d + 1) p n − 1 p d +1 p n 1 wher e µ = ( − 1) m d . Pr o of. see App endix C. R emark . (i). The case (i) a nd (ii) with t = 1 has b een sho wn in [31], Theorem 2. (ii). If k = n/ 6 or 5 n/ 6 , then C 2 has dimension 5 n 0 / 2. Its w eight distribution has b een determined in [15]. 5 App endix A Pr o of of L emma 2 (ii) : F or Y = ( y 1 , · · · , y s ) ∈ F s q 0 , y = y 1 v 1 + · · · + y s v s ∈ F q , w e kno w that F α,β ( X + Y ) − F α,β ( X ) − F α,β ( Y ) = 2 X H α,β Y T (12) is equal to f α,β ( x + y ) − f α,β ( x ) − f α,β ( y ) = T r n d  y p 3 k ( α p 3 k x p 6 k + β p 3 k x p 4 k + β p 2 k x p 2 k + α x )  (13) 20 Let φ α,β ( x ) = α p 3 k x p 6 k + β p 3 k x p 4 k + β p 2 k x p 2 k + α x. (14) Therefore, r α,β = r ⇔ the num ber of common solutions of X H α,β Y T = 0 for all Y ∈ F s q 0 is q s − r 0 , ⇔ the num ber of common solutions of T r n d  y 3 k · φ α,β ( x )  = 0 for all y ∈ F q is q s − r 0 , ⇔ φ α,β ( x ) = 0 has q s − r 0 solutions in F q . Fix an algebraic closure F p ∞ of F p , since the degree o f p 2 k -linearized p oly- nomial φ α,β ( x ) is p 6 k and φ α,β ( x ) = 0 has no m ultiple ro ots in F p ∞ (this fact follo ws from φ ′ α,β ( x ) = α ∈ F ∗ q ), then the zero es of φ α,β ( x ) in F p ∞ , sa y V , form an F p 2 k - v ector space of dimension 3. No t e that gcd( n, 2 k ) = 2 d . Then V ∩ F p n is a v ector space on F p gcd( n, 2 k ) = F p 2 d with dimen sion at most 3 since any elemen ts in F q whic h are linear independen t o v er F p 2 d are also linear indep endent o v er F p 2 k ( see [25], Lemma 4). Since F p 2 d could b e regarded as a 2-dimensional vec tor space ov er F p d , then the p ossible v alues of r α,β is s , s − 2, s − 4 and s − 6 for ( α, β ) ∈ F 2 q \{ (0 , 0) } .  Pr o of of L emma 3 : (i) . W e o bserv e tha t P α,β ∈ F q T ( α, β ) = P α,β ∈ F q P x ∈ F q ζ T r n 1 ( αx p 3 k +1 + β x p k +1 ) p = P x ∈ F q P α ∈ F q ζ T r n 1 ( αx p 3 k +1 ) p P β ∈ F q ζ T r n 1 ( β x p k +1 ) p = q · P α ∈ F q x =0 ζ T r n 1 ( αx p 3 k +1 ) p = p 2 n . (ii). W e can calculate P α,β ∈ F q T ( α, β ) 2 = P x,y ∈ F q P α ∈ F q ζ T r n 1 “ α “ x p 3 k +1 + y p 3 k +1 ”” p P β ∈ F q ζ T r n 1 “ β “ x p k +1 + y p k +1 ”” p = M 2 · p 2 n where M 2 is the nu mber of solutions to the equation ( x p 3 k +1 + y p 3 k +1 = 0 x p k +1 + y p k +1 = 0 (15) If xy = 0 satisfying (15), then x = y = 0. Ot herwise ( x/y ) p 3 k +1 = ( x/y ) p k +1 = − 1 whic h yields that ( x/y ) p 2 k − 1 = 1. Denote b y x = ty . Since gcd(2 k , n ) = d ′ , then t ∈ F ∗ p d ′ . 21 • If d ′ = d , then t ∈ F ∗ p d and (15) is equiv alen t to x 2 + y 2 = 0. Hence t 2 = − 1. There are tw o or none of t ∈ F ∗ p d satisfying t 2 = − 1 dep ending on p d ≡ 1 (mo d 4) or p d ≡ 3 (mo d 4). Therefore M 2 = ( 1 + 2( q − 1) = 2 q − 1 , if p d ≡ 1 (mo d 4) 1 , if p d ≡ 3 (mo d 4) . • If d ′ = 2 d , then by (1 0) w e get (15) is equiv alen t to x p d +1 + y p d +1 = 0. Then w e hav e t p d +1 = − 1 whic h has p d + 1 solutio ns in F ∗ p d ′ . Therefore M 2 = ( p d + 1) ( p n − 1) + 1 = p n + d + p n − p d . (iii). W e ha v e X α,β ∈ F q T ( α, β ) 3 = M 3 · q 2 where M 3 = # n ( x, y , z ) ∈ F 3 q    x p 3 k +1 + y p 3 k +1 + z p 3 k +1 = 0 , x p k +1 + y p k +1 + z p k +1 = 0 o = M 2 + T ′ · ( q − 1) (16) and T ′ is the nu mber of F q -solutions of ( x p 3 k +1 + y p 3 k +1 + 1 = 0 x p k +1 + y p k +1 + 1 = 0 . (17) Canceling y w e ha v e  x p 3 k +1 + 1  p k +1 =  x p k +1 + 1  p 3 k +1 whic h is equiv alen t to ( x p 4 k − x )( x p k − x p 3 k ) = 0 . Therefore x p 4 k = x o r x p k = x p 3 k . • If n/d ≡ 2 (mo d 4), then x ∈ F p 2 d and symmetrically y ∈ F p 2 d . Hence (17) is equiv alent to x p d +1 + y p d +1 + 1 = 0 whic h is the w ell-kno wn Hermitian curv e on F p 2 d . If f o llo ws that T ′ = p 3 d − p d . 22 • If n/d ≡ 0 (mo d 4 ), t hen x ∈ F p 4 d and hence y ∈ F p 4 d . In this case  x p k +1 + y p k +1 + 1  p 3 k = x p 3 k +1 + y p 3 k +1 + 1 and then (17) is equiv a lent to x p d +1 + y p d +1 + 1 = 0 whic h is a minimal curv e o n F p 4 d with gen us 1 2 p d ( p d − 1). Hence T ′ = p 4 d + 1 − p d ( p d − 1) p 2 d − ( p d + 1) = p 3 d − p d . An yw a y , M 3 = ( p n + d + p n − p d ) + ( p n − 1)( p 3 d − p d ) = p n +3 d + p n − p 3 d .  R emark . F or the case d ′ = d , P α,β ∈ F q T ( α, β ) 3 can also b e determined, but we do not need this result. Pr o of of L emma 4 : W e get that q N = P ω ∈ F q P x,y ∈ F q ζ T r n 1 “ ω “ αx p 3 k +1 + β x p k +1 − y p d + y ”” p = q 2 + P ω ∈ F ∗ q P x ∈ F q ζ T r n 1 “ ω “ αx p 3 k +1 + β x p k +1 ”” p P y ∈ F q ζ T r n 1 “ y p d “ ω p d − ω ”” p = q 2 + q P ω ∈ F ∗ q 0 P x ∈ F q ζ T r n 1 “ ω “ αx p 3 k +1 + β x p k +1 ”” p = q 2 + q P ω ∈ F ∗ q 0 P x ∈ F q T ( ω α, ω β ) where the 3-rd equalit y follo ws from that the inner sum is zero unless ω p d − ω = 0, i.e. ω ∈ F q 0 . F o r any ω ∈ F ∗ q 0 , b y (9) w e hav e F ω α,ω β ( X ) = ω · F α,β ( X ), H ω α,ω β = ω · H α,β and r ω α,ω β = r α,β . F rom Lemma 1 (i) w e kno w that T ( ω α, ω β ) = X X ∈ F s q 0 ζ T r d 1 ( X H ω α,ω β X T ) p = η 0 ( ω ) r α,β T ( α, β ) . (18) In the case d ′ = 2 d , b y Lemma 2ii) w e g et that r α,β is ev en. Hence T ( ω α, ω β ) = T ( α, β ) for an y ω ∈ F ∗ q 0 and N = q + ( p d − 1) T ( α , β ).  6 App endix B Pr o of of The o r em 1 (ii): In the case d ′ = 2 d ( n/d is ev en and k /d is o dd) and r α,β = s, s − 2 , s − 4 or s − 6 for ( α, β ) 6 = (0 , 0). According to Lemma 1 and Lemma 5, w e get that for ( α, β ) ∈ N i , T ( α , β ) = ( − 1) m/d + i 2 p m + i 2 d . 23 Com bining L emma 2 and Lemma 3 w e hav e n 0 + n 2 + n 4 + n 6 = p 2 n − 1 (19) n 0 − p d · n 2 + p 2 d · n 4 − p 3 d · n 6 = ( − 1) m/d p m ( p n − 1) (20) n 0 + p 2 d · n 2 + p 4 d · n 4 + p 6 d · n 6 = p n ( p d + 1) ( p n − 1) . (21) n 0 − p 3 d · n 2 + p 6 d · n 4 − p 9 d · n 6 = ( − 1) m/d p m +3 d ( p n − 1) . (22) Solving the system of equations consisting of (19)–(22) yields the result.  Pr o of of The or em 2 : F ro m (1 ) w e kno w that for each non- zero co dew ord c ( α, β ) = ( c 0 , · · · , c l − 1 ) ( l = q − 1 , c i = T r n 1 ( απ ( p 3 k +1) i + β π ( p k +1) i ) , 0 ≤ i ≤ l − 1 , and ( α , β ) ∈ F q × F q ), the Hamming w eigh t of c ( α, β ) is w H ( c ( α, β )) = p n − t ( p t − 1) − 1 p t · R ( α, β ) (23) where R ( α, β ) = X a ∈ F ∗ p t T ( aα, aβ ) = T ( α, β ) X a ∈ F ∗ p t η 0 ( a ) r α,β b y Lemma 1 (i). Let η ′ b e the quadrat ic (m ultiplicativ e) character on F q . Then w e ha v e (1). if d/t or r α,β is ev en, then P a ∈ F ∗ p t η 0 ( a ) r α,β = P a ∈ F ∗ p t 1 = p t − 1 and R ( α, β ) = ( p t − 1) T ( α , β ). (2). if d/t and r α,β are b oth o dd, then P a ∈ F ∗ p t η 0 ( a ) r α,β = P a ∈ F ∗ p t η ′ ( a ) = 0 and R ( α, β ) = 0 . Th us the w eigh t distribution of C 1 can be deriv ed from Theorem 1 and (23) directly . F or example, if d/t is o dd and d ′ = d , then (1). if r α,β = s and T ( α, β ) = p m , then w H ( c ( α, β )) = ( p t − 1)( p n − t − p m − t ). (2). if r α,β = s and T ( α, β ) = − p m , then w H ( c ( α, β )) = ( p t − 1)( p n − t + p m − t ). (3). if r α,β = s − 1, then w H ( c ( α, β )) = ( p t − 1) p n − t . 24 (4). if r α,β = s − 2 and T ( α , β ) = − p m + d , then w H ( c ( α, β )) = ( p t − 1 )( p n − t + p m + d − t ).  7 App endix C Pr o of of L emma 6 : Define n ( α , β , a ) to b e the num ber of γ ∈ F q satisfying (i) and (ii). F rom (9) w e kno w that X H α,β X T = T r n d ( αx p 3 k +1 + β x p k +1 ). Com bining (2), (12) and (13) w e can get 2 X H α,β + A γ = 0 ⇔ 2 X H α,β Y T + A γ Y T = 0 for all Y ∈ F s q 0 ⇔ T r n d ( y φ α,β ( x )) + T r n d ( γ y ) = 0 for all y ∈ F q ⇔ T r n d ( y ( φ α,β ( x ) + γ )) = 0 for a ll y ∈ F q ⇔ φ α,β ( x ) + γ = 0 . (24) Let x 0 , x ′ 0 b e tw o distinct solutions of (i) (if exists). W e can get x 0 = X 0 · V T and x ′ 0 = X ′ 0 · V T with X 0 , X ′ 0 ∈ F s q 0 and V = ( v 1 , · · · , v n ). Define ∆ X 0 = X ′ 0 − X 0 and ∆ x 0 = x ′ 0 − x 0 = X 0 · V T . Then φ α,β ( x 0 ) + γ = φ α,β ( x ′ 0 ) + γ = 0 giv es us 2 X 0 H α,β + A γ = 2 X ′ 0 H α,β + A γ = 0 and hence ∆ X 0 · H α,β = 0 . It follo ws that X ′ 0 · H α,β · X ′ 0 T = ( X 0 + ∆ X 0 ) · H α,β · ( X 0 + ∆ X 0 ) T = X 0 H α,β X T 0 + ∆ X 0 · H α,β · ( ∆ X 0 + 2 X 0 ) = X 0 H α,β X T 0 . Therefore T r m t ( αx ′ 0 p m +1 ) + T r n t ( β x ′ 0 p k +1 ) = T r d t  T r m t ( αx ′ 0 p m +1 ) + T r n t ( β x ′ 0 p k +1 )  = T r d t  X ′ 0 · H α,β · X ′ 0 T  = T r d t  X 0 H α,β X 0 T  = T r d t  T r m t ( αx 0 p m +1 ) + T r n t ( β x 0 p k +1 )  = T r m t ( αx 0 p m +1 ) + T r n t ( β x 0 p k +1 ) . 25 Hence n ( α, β , a ) is w ell-defined (indep enden t of the choice of x 0 ). If (i) is satisfied, that is, φ α,β ( x ) + γ = 0 has solution(s) in F q whic h yields that 2 X H α,β + A γ = 0 has solution(s). Note that rank H α,β = s − i . The refore 2 X H α,β + A γ = 0 has q i 0 = p id solutions with X ∈ F s q 0 whic h is equiv alent to sa ying φ α,β ( x ) + γ = 0 ha s p id solutions in F q . Con v ersely , for an y x 0 ∈ F q , w e can determine γ by γ = − φ α,β ( x 0 ) . Let N ( α, β , a ) b e the n um b er of x 0 ∈ F q satisfying (ii). Then we ha v e n ( α , β , a ) = N ( α, β , a )  p id . Let χ ′ ( x ) = ζ T r t 1 ( x ) p with x ∈ F p t b e an additiv e c haracter on F p t and G ( η ′ , χ ′ ) = P x ∈ F p t η ′ ( x ) χ ′ ( x ) b e the Gaussian sum on F p t . W e can calculate p t · N ( α, β , a ) = P x ∈ F q P ω ∈ F p t ζ T r t 1 “ ω · “ T r n t ( αx p 3 k +1 + β x p k +1 ) − a ”” p = p n + P ω ∈ F ∗ p t T ( ω α, ω β ) ζ − T r t 1 ( aω ) p = p n + T ( α , β ) · P ω ∈ F ∗ p t η 0 ( ω ) s − i χ ′ ( − aω ) where the 3-rd equalit y holds f r om (18) for any ω ∈ F ∗ p t ⊂ F ∗ q 0 . • If s − i and d/t are b oth o dd, and a = 0, then η 0 ( ω ) s − i = η ′ ( ω ) and N ( α , β , 0) = p n − t . • If s − i and d/t are b oth o dd, and a 6 = 0, then N ( α , β , a ) = p n − t + 1 p t · T ( α , β ) · P ω ∈ F ∗ p t η 0 ( ω ) χ ′ ( − aω ) = p n − t + 1 p t · T ( α , β ) · η ′ ( − a ) · G ( η ′ , χ ′ ) = p n − t + ε η ′ ( a ) p n + id − t 2 where the 2- nd equalit y follow s from t he explicit ev a lua tion of quadratic Gaussian sums (see [19], Theorem 5.1 5 and 5.33). • If s − i or d/t is ev en, and a = 0, then η 0 ( ω ) s − i = 1 for an y ω ∈ F ∗ p t and N ( α , β , 0) = p n − t + ε ( p t − 1) p n + id 2 − t . • If s − i or d/t is ev en, and a 6 = 0, then N ( α , β , a ) = p n − t + 1 p t · T ( α , β ) · P ω ∈ F ∗ p t χ ′ ( − aω ) = p n − t − εp n + id 2 − t . 26 Therefore w e complete the pro of b y dividing p id .  Pr o of of The or em 3 (iii): Define Ξ =  ( α, β , γ ) ∈ F 3 q | S ( α, β , γ ) = 0  and ξ =   Ξ   . Recall n i , H α,β , , r α,β , A γ in Section 1 and N i,ε , n i,ε, in the pro of o f Lemma 2. Note that 2 X H 0 , 0 + A γ = 0 is solv able if and only if γ = 0. If ( α , β ) ∈ N i,ε , then the n um b er of γ ∈ F q suc h that 2 X H α,β + A γ = 0 is solv able is q s − i 0 = p n − id . In the case d ′ = 2 d , from Lemma 2 (i) w e get that ξ = p n − 1 + ( p n − p n − 2 d ) n 2 , 1 + ( p n − p n − 4 d ) n 4 , − 1 = ( p n − 1)  1 + ( p 2 n + p 2 n − 9 d − εp 3 m + ε p 3 m − d + ε p 3 m − 3 d − εp 3 m − 5 d − εp 3 m − 7 d + ε p 3 m − 8 d + p n − p n − d − p n − 4 d − p n − 6 d ) / ( p d + 1)  (25) Assume ( α , β ) ∈ N i,ε and φ α,β ( x ) + γ = 0 ha s solution(s) in F q (c ho ose one, sa y x 0 ). Then by Lemma 1 w e g et S ( α , β , γ ) = ζ − T r n 1 „ αx p 3 k +1 0 + β x p k +1 0 « p · T ( α , β ) . Applying Lemma 6 for t = 1 a nd Theorem 1, we get the result.  Pr o of of The or em 4 : Recall M ( α 1 ,β 1 ) , ( α 2 ,β 2 ) ( τ ) defined in (5) and (6). Fix ( α 2 , β 2 ) ∈ F q × F q , when ( α 1 , β 1 ) runs through F q × F q and τ tak es v alue from 0 to q − 2, ( α ′ , β ′ , γ ′ ) runs throug h F q × F q ×  F q  { 1 }  exactly one time. F o r a n y p ossible v a lue κ of S ( α, β , γ ), define s κ = # { ( α, β , γ ) ∈ F q × F q × F q | S ( α, β , γ ) = κ } s 1 κ = #  ( α, β , γ ) ∈ F q × F q × { F q \{ 1 }}   S ( α , β , γ ) = κ  and t κ = # { ( α , β ) ∈ F q × F q | T ( α, β ) = κ } . By Lemma 7 w e ha v e s 1 κ = q − 2 q − 1 × ( s κ − t κ ) + t κ = q − 2 q − 1 × s κ + 1 q − 1 × t κ . 27 Define M κ to b e the n um ber of ( α 1 , β 1 , α 2 , β 2 ) suc h that M ( α 1 ,β 1 ) , ( α 2 ,β 2 ) = κ . Hence w e get M κ = p 2 n · s 1 κ = p 2 n ·  q − 2 q − 1 · s κ + 1 q − 1 · t κ  . Then the result fo llo ws from Theorem 1 and Theorem 3.  Pr o of of The or em 5 : F ro m (1 ) w e kno w that for each non- zero co dew ord c ( α, β , γ ) = ( c 0 , · · · , c q − 2 ) ( c i = T r n t ( απ ( p 3 k +1) i + β π ( p k +1) i + γ π i ) , 0 ≤ i ≤ q − 2 , and ( α, β , γ ) ∈ F q × F 2 q ), the Hamming we ight of c ( α, β , γ ) is w H ( c ( α, β , γ )) = p n − t ( p t − 1) − 1 p t · R ( α, β , γ ) (26) where R ( α, β , γ ) = X ω ∈ F ∗ p t S ( ω α , ω β , ω γ ) . F o r any ω ∈ F ∗ p t ⊂ F ∗ q 0 , w e hav e φ ω α,aβ ( x ) + ω γ = 0 is equiv alent to φ α,β ( x ) + γ = 0. Let x 0 ∈ F q b e a solution of φ α,β ( x ) + γ = 0 (if exist). (1). If φ α,β ( x ) + γ = 0 has solutions in F q , then by L emma 1 and (18 ) we ha v e S ( ω α , ω β , ω γ ) = ζ − „ T r n 1 ( ωαx p 3 k +1 0 + ω β x p k +1 0 ) « p T ( ω α, ω β ) = ζ − „ T r n 1 ( ωαx p 3 k +1 0 + ω β x p k +1 0 ) « p η 0 ( ω ) r α,β T ( α, β ) . Hence R ( α, β , γ ) = T ( α , β ) X ω ∈ F ∗ p t ζ − T r t 1 „ ω · „ T r m t ( αx p m +1 0 )+T r n t ( β x p k +1 0 ) «« p η 0 ( ω ) r α,β . Fix ( α, β ) ∈ N i,ε for ε = ± 1, and supp ose φ α,β ( x ) + γ = 0 is solv able in F q . Denote by ϑ = T r n t ( αx p 3 k +1 0 + β x p k +1 0 ). Then – if s − i and d/ t are b oth o dd, and ϑ = 0, then R ( α, β , γ ) = T ( α , β ) X ω ∈ F ∗ p t η ′ ( ω ) = 0 . 28 – if s − i and d/ t are b oth o dd, and ϑ 6 = 0, then by the result of quadratic Gaussian sums R ( α, β , γ ) = T ( α , β ) η ′ ( − ϑ ) G ( η ′ , χ ′ ) = εη ′ ( ϑ ) p n + id + t 2 , = ( p n + id + t 2 if ε = η ′ ( ϑ ) , − p n + id + t 2 if ε = − η ′ ( ϑ ) . – if s − i or d/t is ev en, a nd ϑ = 0, then η 0 ( ω ) r α,β = 1 f or ω ∈ F ∗ p t and R ( α, β , γ ) = ( p t − 1) T ( α , β ) = ε ( p t − 1) p n + id 2 . – if s − i or d/t is ev en, a nd ϑ 6 = 0, then η 0 ( ω ) r α,β = 1 f or ω ∈ F ∗ p t and R ( α, β , γ ) = − T ( α , β ) = − εp n + id 2 . (2). If φ α,β ( x ) + γ = 0 ha s no solutions in F q whic h implies that φ ω α,ω β ( x ) + ω γ = 0 a lso has no solutions in F q for any ω ∈ F ∗ p t ⊂ F q 0 . Hence S ( ω α , ω β , ω γ ) = 0 and R ( α , β , γ ) = 0. Th us the w eigh t distribution of C 2 can b e deriv ed from Theorem 1, Lemma 6, (25) and (26) directly .  8 Conclus ion and F urther S tudy In this pap er w e ha v e studied the exp o nential sums T ( α , β ) and S ( α , β , γ ) with α, β , γ ∈ F q . After giving the v alue distribution of T ( α, β ) a nd S ( α, β , γ ), w e determin e the correlation distribution among a family of sequence s, and the w eigh t distributions of the cyclic co des C 1 and C 2 . F o r a monomial Dem b o wski-Ostrom function, the asso ciated exp onen tial sums ha v e b een explicitly determined in [2], [3]. F or a general D embows ki-Ostrom function f ( x ), Lemma 1 rev eals the fact that if the num ber of t he solution of the linearized p olynomial relat ed to f ( x ) is explicitly calculated, then the exp onen tial sums P x ∈ F q χ ( f ( x )) and P x ∈ F q χ ( f ( x ) + γ x ) could b e ev aluated explicitly up to ± 1. Thereafter, the correlation distribution of seq uences and the w eight distributions of the asso ciated cyclic co des are also b e determined. In particular, f or the case f ( x ) = x p lk +1 + x p k +1 with l ≥ 5 o dd, w e could get the p ossible v alues of P x ∈ F q χ ( f ( x )) and P x ∈ F q χ ( f ( x ) + γ x ). But the first three 29 momen t iden tities dev elop ed in Lemma 3 is not enough to determine the v alue distribution. Ho w ev er, w e could get the p ossible w eigh ts of the cor r esp o nding cyclic co des. New mac hinery and tec hnique should b e in v en ted t o attac k this problem. 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